Nuprl Lemma : r2-line-eq_weakening
∀l,m:r2-line.  ((l = m ∈ r2-line) 
⇒ r2-line-eq(l;m))
Proof
Definitions occuring in Statement : 
r2-line-eq: r2-line-eq(l;m)
, 
r2-line: r2-line
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
member: t ∈ T
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
r2-line-eq: r2-line-eq(l;m)
, 
and: P ∧ Q
, 
exists: ∃x:A. B[x]
, 
r2-line-sep: r2-line-sep(l;m)
, 
iff: P 
⇐⇒ Q
, 
pi2: snd(t)
, 
pi1: fst(t)
, 
r2-line: r2-line
, 
req: x = y
, 
rev_implies: P 
⇐ Q
, 
nat_plus: ℕ+
, 
le: A ≤ B
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
false: False
, 
top: Top
Lemmas referenced : 
r2-line_wf, 
equal_wf, 
r2-line-eq_wf, 
r2-line-sep_wf, 
int-to-real_wf, 
r2-det_wf, 
rneq-iff, 
nat_plus_properties, 
full-omega-unsat, 
intformand_wf, 
intformle_wf, 
itermVar_wf, 
itermConstant_wf, 
intformless_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_le_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_wf
Rules used in proof : 
hypothesisEquality, 
hypothesis, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut, 
lambdaFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
dependent_functionElimination, 
applyLambdaEquality, 
equalitySymmetry, 
hyp_replacement, 
productElimination, 
independent_functionElimination, 
natural_numberEquality, 
sqequalRule, 
setElimination, 
rename, 
independent_isectElimination, 
approximateComputation, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation
Latex:
\mforall{}l,m:r2-line.    ((l  =  m)  {}\mRightarrow{}  r2-line-eq(l;m))
Date html generated:
2019_10_30-AM-11_32_39
Last ObjectModification:
2018_09_06-PM-03_06_06
Theory : reals!model!euclidean!geometry
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